Glidebending of general caps: an infinitesimal treatment

Author:
Edgar Kann

Journal:
Proc. Amer. Math. Soc. **84** (1982), 247-255

MSC:
Primary 53A05

DOI:
https://doi.org/10.1090/S0002-9939-1982-0637178-X

MathSciNet review:
637178

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Abstract: We prove that two isometric surfaces and with (not necessarily planar) boundary, immersed in , with , having their spherical images in the same open hemisphere, and which are oriented so that the mean curvatures have the same sign, are congruent if they satisfy the glidebending boundary condition:

Let , be the open hemisphere. Then at corresponding boundary points, where and are position vectors of and .

The method uses the fact that the surface is an infinitesimal bending field for the mean surface and is elementary in that it uses the rotation vector of classical infinitesimal bending theory but no integral formulas, maximum principles for elliptic operators or index theorems. The surfaces considered need not have a simple projection on a plane, be convex in the large (an example is given) nor be simply connected. We use the method to prove the finite and infinitesimal rigidity of general caps and caps under glidebending.

**[1]**S. E. Cohn-Vossen,*Verbiegbarkeit von Flächen im Grossen*, Fortschr. Math.**1**(1936), 33-76.**[2]**N. W. Efimow,*Flächenverbiegung im Grossen*, Akademie-Verlag, Berlin, 1957 (German). MR**0105722****[3]**Edgar Kann,*An elementary proof of a finite rigidity problem by infinitesimal rigidity methods*, Proc. Amer. Math. Soc.**60**(1976), 252–258 (1977). MR**0420518**, https://doi.org/10.1090/S0002-9939-1976-0420518-1**[4]**Edgar Kann,*A new method for infinitesimal rigidity of surfaces with 𝐾>0*, J. Differential Geometry**4**(1970), 5–12. MR**0259817****[5]**A. V. Pogorelov,*Extrinsic geometry of convex surfaces*, American Mathematical Society, Providence, R.I., 1973. Translated from the Russian by Israel Program for Scientific Translations; Translations of Mathematical Monographs, Vol. 35. MR**0346714**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1982-0637178-X

Keywords:
Finite rigidity,
infinitesimal rigidity,
glidebending,
bending field,
general cap,
cap,
isometric surface,
congruent surface,
rotation field,
monotypy

Article copyright:
© Copyright 1982
American Mathematical Society